10-Oct-2016: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy
Name: Andrew Martinez
Lab Partner: Richard Mendoza

Statement: To verify that conservation of energy applies to this lab

Introduction: To find an equation for magnetic potential energy. To find the magnetic potential energy for a non-constant potential energy one must recognize the relationship between potential energy and force. The relationship appears as U(r)= - (integral ^r _infinity) F(r)dr.

Apparatus: Air track (Friction less surface), glider, magnets, air pump, books

Books were used to raise the air track to multiple angles to plot the separation distance

Procedure
- Level the air track to measure the variable h.
- From h this will be the position the experiment is done at.
- Collect data from various angles to plot a relationship between magnetic force F and the separation distance r.
- Plot F vs r graph. The relationship in this graph is a power law F = Ar^n. Find A and n from the curve fit performed on the graph
- Verify conservation of energy
- Attach an aluminum reflector
- Weigh the cart
- Determine the relationship between the distance the motion detector reads and the separation distance between the magnets
- Make a single graph showing KE, PE, and total energy of the system as a function of time.

Measured Data:
Using books we take the angle of the air track at different points and measuring the separation distance between the glider and the magnet. (+/- .1 degrees, +/- .2 mm).
1. .6 degrees, 22.2 mm
2. 2.1 degrees, 18.2 mm
3. 4.0 degrees, 14.1 mm
4. 5.2 degrees, 12.9 mm
5. 8.3 degrees, 10.6 mm
Convert mm to meters

Using this data we can use the equation F_mag=sin(theta)mg to find the force of the magnet
1. .03456 N
2. .210554 N
3. .2304 N
4. .299 N
5. .477 N

Graphed Data:
Using the measure data we plot a Distance(r)(x-axis) vs. Forces(F)(y-axis)

Using graph we perform a curve fit to find the A=2.363*10^-6 (+/- 1.368*10^-6) and n=-2.690 (+/- 0.1301) to plug into a power law (F=Ar^n)

Using the graphs we can verify that conservation of energy applies to this system

The bottom graph represents the single graph of KE, PE, and total energy as a function of time. As can be seen the graphs come together at a single point.

Conclusion: Despite knowing that conservation of momentum does apply to this experiment based on our graphs results it can not be proven with the data provided. Their are a number of reasons why the graphs fails to show the conservation of energy but one must likely answer is due to systematic error where the human element may have miscalculated or used a wrong equation. 

14-Oct-2016: Work-Kinetic Energy Theorem Lab

Lab 11: Work-Kinetic Energy Theorem
Name: Andrew Martinez,
Lab Partner: Richard Mendoza, Lynel Ornedo, Mohammed Karim

Statement: In experiment 1 the object is to calculate work done by finding the area under the graph. In experiment 2 the objective is to find the spring constant.

Introduction: This lab is divided into two parts called EXPT 1: Work Done by a Non-constant Spring Force and EXPT 2: Kinetic Energy and the Work-Kinetic Energy Principle. The main objective of EXPT 1 is to find the work done by force by using the area and EXPT 2 is to find the spring constant by using graphs.

Apparatus

Cart attached to a spring and placed upon a long horizontal surface. A force probe is placed on the other end of the track to record the force over the distance traveled.

EXPT 1: Work Done by a Nonconstant Spring Force

Procedure
- Calibrate the force probe with a force of 4.9 applied
- Set up the apparatus
- Ensure that the motion detector is able to detect the cart.
-  Open up the file L11E2-2 and zero the force probe. Then begin graphing the force vs position graph as the cart is pushed forced to about .6 m
- Determine the spring constant and explain how
- Use integration routine in the software to find work done by the stretch of the spring.

Data recorded by the software as the cart is slowly pushed towards the force probe. The force probe should be set to reverse direction to plot in the positive x direction. We then use a linear fit to find the spring constant which should be our slope. Slope is 8.828 N/m which is the spring constant. We know that this is true because of hooke's law that states a spring force is equal to the spring constant times its position or F=k*x.

When then use the integration routine to find the work done by stretching the spring which is shown to be .3930 Nm W=F*d or work equals force times displacement but in this case we are using the area under the graph to find the work done by the spring. 

EXPT 2: Kinetic Energy and The Work-Kinetic Energy Principle

Procedure
- Measure the mass of the cart m=.6997 kg
- In new calculated column use a formula to find kinetic energy of the cart at any given point
- Repeat steps in previous experiment for setup
-  Find the change in kinetic energy after the cart is released from the initial to several different positions
- Find the work done by the spring up to those differing positions

Graph Analysis

As shown by the graphs we can find the work done at different positions as well as the kinetic energy

Conclusion: The work done on the cart by the spring shows how kinetic energy changes in relation to work. Since we know that F=k*x and that W=F*d we can see how the change in the kinetic energy of an object is equal to the net work done on the object and thus proves the work-kinetic energy principle.

3-Oct-2016: Centripetal Force with a Motor

Lab 9 Centripetal Force with a Motor
Date: 03 October 2016
Name: Andrew Martinez
Lab Partners: Richard Mendoza, Lynel Ornedo, Mohammed Karim

Statement: To determine the relationship between theta and omega.

Introduction: As angular speed increase due to the motor spin the mass shifts to a larger radius resulting in an increase in angle theta.

Apparatus:

An electric motor mounted on a surveying tripod. Long string tied to the end of the horizontal rod. Rubber stopper at the end of the string.

Long shaft going vertically up from shaft with a horizontal rod mounted on the vertical rod. Ring stand with a horizontal piece of paper on tape sticking out.

Procedure
- The professor has control over the motor and thus can increase it for the experiment.
- The students use the vertical stand that with the paper sticking out to find the mass height from the ground.
- Students are also responsible for measuring the amount of time it takes for the mass to make ten rotations
- Using all the measured data students can then conceptually break down the components of the centripetal force to derive an equation for omega
- Using your actual and calculated values one can then graph these omegas to how close the correlation is and the percent error to see if theta is related to omega.

Measured Data

Using our understanding of centripetal forces we are able to derive equations and variables based upon this understanding. Through this we can display the relationship between θ and ω.

On this page we derive several equations to help in proving the relationship between theta and omega. H is a constant as it stands for the vertical stands height. Little h is the shifting height of the swinging mass as the motor increases in speed. L is the length of the string and while the swinging mass may shift height L will always maintain its length. Knowing this we can find theta through SOH CAH TOA. Since only A(H-h) and H(L) are known we can use the arccosine to find theta. Since we know that omega is equal to theta divided by time we can find the actual omega.

In this picture we are trying to derive an equation to find omega by calculation and understanding of centripetal force. We break our net centripetal force into its X and Y components where Fx: Tsinθ=mrω^2, Fy: Tcosθ-mg=0. By setting T=mg/cosθ we can substitute into T to solve for omega as seen above where the masses cancel each other out, sin over cos can be changed into tan and r is r+Lsinθ to account for the changing radius.

 Graphing Analysis

By graphing our actual values of omega and the calculated value of omega and performing a linear fit with a correlation we can see that the correlation is .9999 almost at 1. 

Upon conducting a percent error test we can see that the actual and calculated values are reasonable close to each other and that by doing so we can see that their is a relationship between omega and theta.

 Conclusion: Based on our results we can conclude that there is a relationship between omega and theta. For as theta increase so did omega and that by deriving an equation for omega through trigonometry and using theta we were able to get accurate results for omega.  Based upon our low percent errors it is safe to presume that theta is related to omega. One concern that should be noted in the rising percent error as the values went further along. Some possibilities for this could be a random error of changing speed in the motor or a systematic error of timing of the rotations.

28-Sept-2016: Centripetal Force Lab

Lab 8: Centripetal Force with changing mass, radius, and angular velocity
Name: Andrew Martinez
Lab Partner: Professor Phillip Wolf

Purpose: To determine the relationship between centripetal force and mass, centripetal force and radius, and centripetal force and angular velocity.

Theory/Introduction: To find centripetal force the formula F_centripetal = mrw^2 is used where w is the angular velocity, r is the radius, and m is the mass. By varying the values on m, r, and w separately and then graphing in a centripetal force vs (varying variable) our slope will equal our constant values and prove their relationship to one another and to the formula.

Apparatus: Was a custom made motor that spun in a circular motion. A time gate with some mass was fasted to the spinning panel and measured for the varying variables.

Measured Data

Trail 1: M=200 g, R= 19 cm, V=6.1 Volts, Delta Theta = 20pi, Delta T=.43s-20.63s, F=0.392 N
Trail 2: M=200 g, R= 28.5 cm, V= 6.1 V, Delta Theta= 20pi, Delta T=1.27s-19.28, F=0.837 N
Trail 3: M=200 g, R=40 cm, V=6.1 V, Delta Theta= 20pi, Delta T=.55s-16.7s, F=1.39 N
Trail 4: M=200 g, R=54 cm, V=6.1 V, Delta Theta= 20pi, Delta T=.94s-17.47s F=1.65 N
Trail 5: M=200 g, R=54 cm, V=6.6 V, Delta Theta=20pi, Delta T=.2s-13.1s F= 2.61 N
Trail 6: M=200 g, R=54 cm, V=7 V, Delta Theta= 20pi, Delta T=.7s-12.5s, F=3.37 N
Trail 7: M=200 g, R=54 cm, V=7.7 V Delta Theta=20pi, Delta T=.32s-10.04s, F=4.79 N
Trail 8: M=100 g, R=54 cm, V=7.7 V, Delta Theta=20pi, Delta T=.6s-10.4s, F=2.3 N
Trail 9: M=50 g, R=54 cm, V= 7.7 V, Delta Theta=20pi, Delta T=.31s-10.54s, F=1.14 N

Graphing Analysis

In this Force vs Rw^2 graph the variable m for mass is kept constant. By comparing the slope to the actual M one can see how accurate the equation F=mrw^2 is.  In this case the slope of mass is 0.21 when our actual mass is 0.2 which means we had a 5% percent error which is an acceptable margin of error.

In this Force vs w^2 graph the variable mr for mass and radius is kept constant. By comparing the slope to the actual mr one can see how accurate the equation F=mrw^2 is.  In this case the slope of mass and radius is 0.117 when our actual is 0.108 which means we had a 8.3% percent error which is admittedly pushing the acceptable margin of error.

In this Force vs mw^2 graph the variable r for radius is kept constant. By comparing the slope to the actual r one can see how accurate the equation F=mrw^2 is.  In this case the slope of radius is 0.393 when our actual radius is .54 which means we had a 27.22% percent error which is too large of an error to be consider acceptable.

Conclusions:

Based on the data and graphing analysis we can conclude that the equation F=mrw^2 does indeed work. While both graphs Force vs w^2 and Force vs Rw^2 slope and actual were within acceptable percent error range the graph Force vs Mw^2 was not. This can be contributed to a number of things from random errors such as changes in voltage or the friction on the mass as it was accelerated to systematic errors suchs as inaccurate measurements or timing. If the lab were to be redone I am confident that all three graphs of data can be brought into reasonable range of percent error.

12-Oct-2016: Ballistic Pendulum

Lab: Ballistic Pendulum
Name: Andrew Martinez
Lab Partners: Richard Mendoza, Lynel Ornedo

Statement: To determine the firing speed of a ball in the spring-loaded experiment

Introduction: In this lab we will be for three things. The firing speed of the ball and to determine the trajectory the ball will travel and its landing point. Using our knowledge of potential energies, conservation of momentum, and conservation of energy we can determine all three.

Procedure:
Part 1
- Measure the length of the string holding the block in place
- Measure the height of the table
- Record the masses of the ball and block
- Place the ball into the spring-loaded gun and select which level to set it at
- Place the angle indicator at zero
- Fire the ball into the block and record the max angle
- Repeat five times
- Use the conservation of momentum to write an equation to find the speed of the system after the collision

Part 2
- Use the value gained in part one after the collision
- Use conservation of energy to and relate the max height to the initial speed of the block
- Using these value predict where the ball well land if launch from table height then find actual

Apparatus:

Ballistic Pendulum with the scale to measure mass

Measured Data
Mass_ball: 7.6 g +/- .1 g
Mass_block: 79.2 g +/- .1 g
Tables height: 85.8 cm +/- .01 cm
String length: 20 cm +/- .01 cm
Trail Angles: 17 degrees, 15 degrees, 16.5 degrees, 17 degrees, 14.5 degrees
Average Angle: (17+15+16.5+17+14.5)/5 = 16 degrees

Calculated work

Vf of mass (ball+block) = .3897 m/s
Vi of mass ball = 4.45 m/s
T = .436 s
X = 1.94 m (experimental)
X = 2.02 m (Actual)

Analysis:
Setting kinetic energy = to gravitational potential energy we get an equation which can be rearrange to solve for Vf = .3897 m/s. Using the value of Vf we relate the height to initial velocity and solve for Vi getting 4.45 m/s. We then use kinematic equations for the x-axis to find how far the ball will travel before impact and use a kinematic for the known height to find time. We get 1.94 m as the estimate range of impact. Upon actually launching we found the true impact point to be 2.02 m.

Conclusion:
This lab demonstrates how by using the laws of conservation of momentum and energy as well potential energies one can determine unknown velocities and predict there range. These estimations are reasonable accurate with a found percent error of 3.96%.

21-Sept-2016: Modeling Friction Forces

Lab 7 Modeling Friction Forces
Date: 21 September 2016
Lab Partner: Richard Mendoza, Mohammed Karim, Lynel Ornedo
Name: Andrew Martinez

Introduction: Conducting five different experiments that involve static and kinetic friction.

Apparatus:
- Wooden blocks
- Various mass plates
- String
- Wooden Panel
- Hanging mass
- Computer
- Ring Stand
- Focus Sensor
- LabPro
- Clamps
- Motion Detector

Procedure
Part 1: Static Friction
- Set up apparatus on uniform surface as shown adding weights for each block until it moves.
- Record the mass of each block until reach 4 and the weight it moved on
- Plot data into computer

Apparatus for part 1 with maximum number of blocks

Hanging Mass and pulley along with the string for tension

Blocks with tension from string

Part 2: Kinetic Friction
- Calibrate the force sensor
- Plug force sensor into LabPro and connect to computer via USB. Set to 10-N range
- Use 500-gram for calibration
- Find the mass of one of the wooden blocks
- Collect data from force sensor by moving the block horizontally at a constant speed along a uniform surface as LabPro
- Repeat till the last block
- Plot data and the slope gives the kinetic friction

Example of calibration of the force sensor

Part 3: Static Friction From a Sloped Surface
- Place the wood panel at an incline angle while leaving a block on it
- Raise until the block begins to move downward on its on
- Take the angle of that moment in order to help determine the coefficient of static friction

Part 4: Kinetic Friction From Sliding a Block Down an Incline
- Attach a motion sensor just above the wooden panel
- Leaving the Panel at the same incline find the coefficient of kinetic friction by using the motion sensor to find the acceleration.

Used for Part 3 & 4 to determine the static and kinetic friction. On top of the board is the motion sensor.

Part 5: Predicting the Acceleration of a Two-Mass System
- Use the kinetic friction you got from part 4 and derive an expression for the acceleration of a block with a hanging mass

Data:
Part 1
Block 1: 179.1 g
Block 2: 177.5 g
Block 3: 181.7 g
Block 4: 190.1 g
Hanging Mass
B1: 85 g
B1B2: 150 g

B1B2B3: 200 g
B1B2B3B4: 310 g

X:Normal Force vs Y:Force of static friction
By listing the normal force with its force of static friction we get our slope

Slope is .4106 making the coefficient of static .4106

Part 2

Using the data given by these charts plug there values into the Y for the graph below

With the Y from the graphs on top we can find the kinetic friction from the slope.
Uk = .2300

Part 3

Using the angel we found the block to move out we can determine the coefficient of static friction by substitution and plugging in.
Us = .4306

Part 4

Using the angel we found the block to move at and continue to move we can determine the coefficient of kinetic friction by substitution and free body diagram.
Uk = .2295

Part 5

Used Block 1 mass: 179.1 g hanging mass at B1: 85 g
Experimental a=2.068 m/s^2

Using the motion sensor then recording the acceleration the block moves out at the given masses.
Actual a(slope) = 1.682

Percent error: 22.94%

Conclusion: Aside from part 5 the experiments went according to plan. Part 5 has a large percent error due to human or systematic error. There could have been mistakes in which block was used or the pulley system was properly put in place or the motion sensor might have been placed to far. Whatever the case the majority of the experiment was a success and if part 5 were to be redone I am sure it would achieve more favorable results.

7-Sept-2016: Propagated uncertainty in measurements

Lab 6 Propagated Uncertainty in Measurements
Date: 7 September 2016
Lab Partner: Richard Mendoza
Name: Andrew Martinez

Introduction: Use propagated uncertainty to find the propagated error in the density of the metal cylinders

Apparatus:
- Iron Cylinder
- Aluminum Cylinder
- Calipers
- Mass scale

Procedure: Using the scale and caliper find the mass, height, and diameter of the cylinders as well as there expected error and use these measurements to determine density with propagated error.

Caliper used to find the length and diameter to an .01 expected error

Aluminum Cylinder

Iron Cylinder

Mass Scale with an .1 expected error

Measure Data
Aluminum Cylinder: M(29.1 g +/- .1 g), H(7.83 cm +/- .01 cm), D(1.29 cm +/- .01 cm)
Iron Cylinder: M(30.2 g +/- .1 g), H(3.07 cm +/- .01 cm), D(1.24 cm +/- .01 cm)

Picture shows the work done to find the density and its propagated error

Density for aluminum is 2.84 g/cm^3 +/- .288 g/cm^3
Density for iron is 8.14 g/cm^3 +/- .826 g/cm^3

Conclusion: The experimental measurements with their expected errors yielded fairly accurate results for density and the propagated error. Random errors that may have occurred is misreading on the measuring scales. Systematic errors might be user resulted in the calculations of the math problems.

14-Sept-2016: Trajectories

Lab 5 Trajectories
Date: 14 September 2016
Lab Partner: Richard Mendoza
Name: Andrew Martinez

Introduction: Using your understanding of projectile motion estimate where the ball will impact on an inclined board.

Apparatus:
- Aluminum "v-channel"
- Steel ball
- Board
- Ring Stand
- Clamp
- Paper
- Carbon paper

The launcher without the board. You can see the ring stand, aluminum "v-channel", clamp, and steel ball.
V-channel is placed at an incline to cause downward acceleration.

Launcher with board. Wood board placed at a 19.2 degree angle, carbon paper on board, and paper underneath carbon paper.

Another view of the inclined view of the apparatus. 

Procedure: Set up apparatus without wooden board. Determine height of the ramp from the floor and incline. Let the steel ball launch from the ramp and determine its range. Does this five times with carbon paper placed under the assumed impact point. Find the distance for each and find the overall average in distance. Place wooden board on table directly underneath the ramp launch point. Using the information learned determine the expected landing point on the wooden table and then place carbon paper on the wooden ramp to measure the actual.

Measured Data
h=94.2 cm
Theta=19.2
D1=108.7 cm, D2=108.1 cm, D3=106.4 cm, D4=106.1 cm, D5=105.8 cm

Finding Velocity
V=242.5 cm/s

Velocity was determined by using the distance, height, and gravity.

Finding Impact Point
Use the angle and velocity found earlier to determine impact point
Estimated distance is 44.25 cm

Distance was determined by using the found angle and velocity as well as subtitution to account for unknowns

Actual distance is 44.6 cm

Conclusion: The percentage error between the approximated and the actual was found to be .78%. This makes the calculations to be extremely accurate. One systematic error would be at what height the steel ball was released on the steel ramp. This could give some increase acceleration at different points.

19-Sept-2016: Modeling the fall of an object falling with air resistance

Lab 4: Modeling the fall of an object falling with air resistance
Date: 14 & 19 September 2016
Lab Partner: Richard Mendoza
Name: Andrew Martinez

Introduction: Air resistance applied to an object depending on its speed, shape, and the material. This model can be seen as a power law.

Apparatus:
- 150 coffee filters (mass= 134.2 g)
- Meter stick
- Computer

Procedure:
- Take the coffee filters and measure their mass
- Find a high balcony inside a building to minimize air resistance
- Bring laptop to record with logger pro
- Use a meter stick for a point of reference for the recording
- Drop the filters 5 times recording each instance
- Using logger pro graph each video and find the liner fit to plug into part 2 of excel
- Use excel to find terminal velocity

Part 1 Data
The five graphs seen here are position vs time graphs of each of the five videos. The data was taken by scaling the one meter mark and using dots to follow the distance the filters had moved per frame during free fall. This was then calculated by the computer and put into graph form.

Video 1

Terminal Velocity (Slope): 3.088 m/s

Video 2

Terminal Velocity (Slope): 2.156 m/s
Video 3

Terminal Velocity (Slope): 2.842 m/s
Video 4

Terminal Velocity (Slope): 3.373 m/s
Video 5

Terminal Velocity (Slope): 3.100 m/s
Using a power law Fresistance = kv^n (k=a, b=n)
Terminal Velocities Graph

K= .0001901 +/- 0.0002173
N = 4.476 +/- 0.9853

Part 2
Measured Data
Excel spreadsheet to model the fall of an object with air resistance using the terminal velocities.

The predicted terminal velocity for various coffee filters appears to be 5.5 m/s
Plug it into power law Fresistance = kv^n. 0.0001901*(5.5^4.476) = .39 N

Conclusion: K and N are determined from the linear fits done to the video graphs and the results that appear of the terminal velocity graph. There is an uncertainty in terms of how accurate the graphs are depending on the user accuracy on plotting the dots. Air resistance was found to be .39 N and based on the model it was effective in determining the air resistance. Other might like to use this method because it requires less analytical work.